Driving Hair from a Texture
A sleepless night brought up the following question: How to add a Particle Texture?

The steps are finicky but work well:


Create the particle system, and create/paint a new texture; include it as the object texture if you want to see what you're doing (the object's texture/material won't affect the particle system).
In the Particle Properties tab there's a dropdown called Textures. Select your texture.
In the Texture Properties tab, go under the Image > Settings dropdown and select your image (again).
At the very top of the Texture Properties tab, set the dropdown to ParticleSettings - Texture.
Finally, under Influence (still in Texture Properties), check Density to let the image drive particle density.


Also, for the particles to appear at the right place on a non-square surface, it may be necessary to select the UV unwrap for the particle system to match the texture placement. That's under Texture Properties > Mapping > Coordinates, select UV. 

Steps 3 and 4 are particularly strange at first, but work well after all. Tabbing in/out of edit mode will reflect the particle system changes. Then, to texture paint the hair, go to texture paint mode and paint more white on the texture, remembering to tab in/out of texture paint mode to update the particle system.